In ring theory, a branch of mathematics, the radical of an ideal of a commutative ring is another ideal defined by the property that an element is in the radical if and only if some power of is in . Taking the radical of an ideal is called radicalization. A radical ideal (or semiprime ideal) is an ideal that is equal to its radical. The radical of a primary ideal is a prime ideal.

This concept is generalized to non-commutative rings in the Semiprime ring article.


The radical of an ideal in a commutative ring , denoted by or , is defined as

(note that ). Intuitively, is obtained by taking all roots of elements of within the ring . Equivalently, is the preimage of the ideal of nilpotent elements (the nilradical) of the quotient ring (via the natural map ). The latter proves that is an ideal.[Note 1]

If the radical of is finitely generated, then some power of is contained in .[1] In particular, if and are ideals of a Noetherian ring, then and have the same radical if and only if contains some power of and contains some power of .

If an ideal coincides with its own radical, then is called a radical ideal or semiprime ideal.



This section will continue the convention that I is an ideal of a commutative ring :


The primary motivation in studying radicals is Hilbert's Nullstellensatz in commutative algebra. One version of this celebrated theorem states that for any ideal in the polynomial ring over an algebraically closed field , one has



Geometrically, this says that if a variety is cut out by the polynomial equations , then the only other polynomials which vanish on are those in the radical of the ideal .

Another way of putting it: the composition is a closure operator on the set of ideals of a ring.

See also


  1. ^ Here is a direct proof that is an ideal. Start with with some powers . To show that , we use the binomial theorem (which holds for any commutative ring):
    For each , we have either or . Thus, in each term , one of the exponents will be large enough to make that factor lie in . Since any element of times an element of lies in (as is an ideal), this term lies in . Hence , and so . To finish checking that the radical is an ideal, take with , and any . Then , so . Thus the radical is an ideal.
  2. ^ For a proof, see the characterisation of the nilradical of a ring.
  3. ^ This fact is also known as fourth isomorphism theorem.
  4. ^ Proof: implies .


  1. ^ Atiyah–MacDonald 1969, Proposition 7.14
  2. ^ Aluffi, Paolo (2009). Algebra: Chapter 0. AMS. p. 142. ISBN 978-0-8218-4781-7.
  3. ^ Atiyah–MacDonald 1969, Proposition 4.2
  4. ^ Lang 2002, Ch X, Proposition 2.10